Most directors would be gratified to know that tickets for their show are the most coveted in London. Michael Grandage, however, is suffused with guilt. His sparse, heart-wrenching production of Othello, starring Chiwetel Ejiofor and Ewan McGregor, sold out months ago. "I've tried to persuade the company to transfer," he says of the production at the Donmar Warehouse, which he runs. "But they've all got projects lined up back-to-back. All I can say is sorry, really sorry." His earnest manner, quite unlike his usual ebullience, suggests his apology to those missing out is heartfelt.

Grandage does, however, have something to offer in compensation: booking opens today for a star-studded new Donmar season featuring Kenneth Branagh, Judi Dench, Derek Jacobi and - the big draw for many - Jude Law, playing Hamlet. And the venue won't be the 250-seat Donmar, which will run its own programme (still to be announced) concurrently - but the much more capacious Wyndham's Theatre in the West End. Grandage makes a good show of trepidation: "It's 750 seats; none of us has any idea what's going to happen." But chances are these tickets will prove pretty desirable, not least because they're selling at Donmar prices - significantly lower than the West End average.

For such a bijou space, the Donmar is surprisingly vital, displaying an ambition and scope to rival the National. Under Grandage's predecessor, Sam Mendes, it radiated an aura of glitz, especially when Mendes lured Hollywood stars such as Nicole Kidman and Gwyneth Paltrow to perform there. Grandage, though, has transformed the Donmar into "a big house of ideas", focusing on lesser-known plays and European writers, work that is cerebral and emotionally challenging, with a passion that has proved, for audiences, unexpectedly exciting and enticing. He will be doing the same in the West End, with a rarely performed Chekhov and a play by the Japanese writer Yukio Mishima - the kind of thing most West End theatres, even some fringe ones, would consider box-office poison.

After five years in the job, the effusive 45-year-old shows no sign of restlessness. He still has plenty of ambition for the place: to transfer work to New York, to programme ever more unusual plays, to take more productions on British tours. He also recently signed up to direct his first opera, at Glyndebourne in 2010, "but that seems like aeons away". He's too focused on the present to care much what the future may hold.

Similarly, he won't dwell on the past. "I'm ruthless about that," he says. "Once it's behind you, you just move on." It's not that he's rejected his past - he spends most of his free time in Cornwall, where he grew up. It's just that there's nothing sentimental for him about these visits: "The past that was young when I was a kid is now either dying or dead." Recently, he dreamed that on one of these excursions home he encountered himself as a 10-year-old. "I didn't recognise myself," he says. "I mean, I did and I thought, 'Ugh, weird little kid.'" There was nothing of himself as he is now, except "a bit of creativity" that was yet to be channelled. That he has since become a successful director he puts down to "a lot of luck" and "key individuals at key moments".

Among them were a couple who ran an amateur dramatics group in Penzance, who noticed that he was keen on acting, and a music teacher "who was more of a life teacher", instructing him to describe his likes and dislikes and so help "build taste". His father, who ran a sweetshop but painted in his free time, augmented this by "constantly asking me about why a painting was good or bad". Then there was his first flatmate when he moved to London aged 18 to go to drama school, who took him to operas and ballets and urged him "to have an opinion" on what he'd seen.

He makes himself sound shy, needing the encouragement of others to express himself. Far from it, says Grandage: "Even at a relatively early age, I liked acting, so somewhere in there I must have enjoyed showing off. I was the last head boy in the history of the Humphry Davy grammar school, before it became a comprehensive, and the headmaster told me it was because I was 'good at giving a speech'. I thought, 'Oh God: not because I'm academic? Good at sports?' He knew lots of speeches would need to be made, and I got the job."

Plenty of proper acting jobs followed Grandage's graduation from the Central School of Speech and Drama in 1984, though you wouldn't know it from his current CV: his 12-year acting career isn't mentioned. He simply wasn't happy as an actor, and it's the thing he has been most ruthless about forgetting. Intriguingly, though, some of his greatest successes as a director, notably Othello, and his riveting production of Don Carlos in the West End in 2005, have been with plays he once acted in, plays he thinks he was "too young to appreciate".

He made the switch to directing in 1995, choosing for his first production Arthur Miller's The Last Yankee because, with only four actors and essentially one act, it wouldn't overwhelm him with technical difficulties. He enjoyed himself so much that he instantly abandoned acting - though that may have had something to do with the fact that he met his partner, the designer Christopher Oram, on that production. Their first meeting, says Grandage, was "perfectly lovely, but not significant". It was at their second encounter, when Oram showed Grandage his model of the set - which was precisely what Grandage had imagined, only "20 times more beautiful" - that Grandage fell in love. "There is something wonderful about having a soulmate," he says, "who shares the same approach to noticing why something is exciting; someone who understands what in the world is right and good and perfect, and why we would like to achieve elements of that together, and more importantly why we just want to experience it together. That's a lovely, precious thing that matters more than anything."

Apart from Oram, Grandage has built up a team of regular collaborators. He has also, from an early stage, shown a marked preference for running a theatre, or even theatres, rather than working freelance, so much so that his first two years as artistic director of the Donmar were also his last of a five-year stint running Sheffield Theatres, a period during which "people thought I was insane, but for me was brilliant". It's tempting to speculate that he seeks this security as a kind of reaction to the death of his father in an accident when he was 18, but Grandage - while not ruling that out - says there's another, simpler reason.

"One of the joys of running your own building is that you can direct plays. If I were freelance, I'd have to go off to every theatre in the country saying, 'I've got a good idea for a play.' If I have a good idea for a play here, I can do it. And if people think I'm hogging the Donmar, they can frankly jump, because that's the reward for going out and raising money and doing all the other parts of the job."

That isn't to say he doesn't enjoy the administrative elements of his work. "I love sitting in front of the computer with a lot of books, seeing how things look together, then coming in here and saying, 'Right, here's what we should do,' and making it happen. That's the thrilling bit." All the more so as he learns "not to try and please all the people all the time - or even please anybody. I'm still working on the idea that you programme a theatre from your own enthusiasms."

If that enthusiasm ever abates, that's when he'll address the future: "I know myself well enough now that I'll give up if I stop getting excited about it".

· Othello is at the Donmar Warehouse, London, until February 23. Tickets for the Donmar West End season go on sale today. Box office: 0870 060 6624.

Sign up for the Guardian Today

Our editors' picks for the day's top news and commentary delivered to your inbox each morning.